On the Smallest Positive Singular Value of a Singular M-Matrix with Applications to Ergodic Markov Chains

Abstract

Let A be a singular $n \times n$M-matrix of rank $n - 1$ and let B be an $( n - 1 ) \times ( n - 1 )$ nonsingular matrix that results from deleting row j and column k from A. An open question discussed by Harrod and Plemmons [SIAM J. Sci. Stat. Comput., 5 (1984), pp. 453–469] is whether there is a choice of j and k such that $\sigma _{n - 1}^{( A )} \approx \sigma _{n - 1}^{( B )} $ where $\sigma _{n - 1}^{( A )} $ is the smallest positive singular value of A and $\sigma _{n - 1}^{( B )} $ is the smallest singular value of B. In this paper, we resolve this conjecture by showing that there is always such a choice. This conjecture is important when finding the stationary distribution of an ergodic Markov chain. This can be posed as the problem of finding the $n \times 1$ vector p such that $Ap = 0$ and $e^T p = 1$ where $e = ( 1, \cdots ,1 )^T $. Here $A = I - Q^T $ where Q is a row stochastic matrix and p is the stationary distribution vector of the chain. This problem can be reduced to the problem of solving a system of linear equations with coefficient matrix B. If $\sigma _{n - 1}^{( A )} \approx \sigma _{n - 1}^{( B )} $ then this system of linear equations is about as well conditioned as the original problem.